International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. B. ch. 3.3, p. 373   | 1 | 2 |

Section 3.3.1.3.11. Other useful rotations

R. Diamonda*

aMRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England
Correspondence e-mail: rd10@cam.ac.uk

3.3.1.3.11. Other useful rotations

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If rotations in display space are to be controlled by trackerball or tablet then there are two measures available, an x and a y, which can define an axis of rotation in the plane of the screen and an angle θ. If x and y are suitably scaled coordinates of a pen on a tablet then the rotation [\pmatrix{\displaystyle{y^{2} + x^{2}c\over x^{2} + y^{2}} &\displaystyle{-xy(1 - c)\over x^{2} + y^{2}} &x\sqrt{x^{2} + y^{2}}\cr \noalign{\vskip3pt} \displaystyle{-xy(1 - c)\over x^{2} + y^{2}} &\displaystyle{x^{2} + y^{2}c\over x^{2} + y^{2}} &y\sqrt{x^{2} + y^{2}}\cr \noalign{\vskip3pt} -x\sqrt{x^{2} + y^{2}} &-y\sqrt{x^{2} + y^{2}} &c\cr}] with [c = \sqrt{1 - (x^{2} + y^{2}){^2}}] is about an axis in the xy plane (i.e. the screen face) normal to [(x,y)] and with [\sin \theta = x^{2} + y^{2}]. Applied repetitively this gives a quadratic velocity characteristic. Similarly, if an atom at [(x, y, z, w)] in display space is to be brought onto the z axis by a rotation with its axis in the xy plane the necessary matrix, in homogeneous form, is [\pmatrix{\displaystyle{x^{2}z + y^{2}r\over x^{2} + y^{2}} &\displaystyle{-xy(r - z)\over x^{2} + y^{2}} &-x &0\cr \noalign{\vskip3pt} \displaystyle{-xy(r - z)\over x^{2} + y^{2}} &\displaystyle{x^{2}r + y^{2}z\over x^{2} + y^{2}} &-y &0\cr\noalign{\vskip3pt} x &y &z &0\cr 0 &0 &0 &r\cr}] with [r = \sqrt{x^{2} + y^{2} + z^{2}}].








































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